However much society has changed since the 1950s, we are still locked in a world where women are presumed to be the prime carers of children. Photograph: Alamy

Picture the scene: two opposing senior politicians are desperately juggling the demands of work and family. The frantic Sunday morning interview schedule means that while the opposition spokesperson delivers a radio interview savaging the government's economic policy, the minister has to take over babysitting duties in the green room.

A fictional satire of the travails of female MPs? A vision of life in the new Icelandic matriarchy? No. According to the Daily Mirror, it is Britain today; a sketch of modern fatherhood played out with Ed Balls and George Osborne in the leading roles. It takes a Westminster village to raise a child, or so it seems.

Before your cynical old heart melts, a quick reality check. According to my extensive investigative journalism (I Googled it), Ed Balls's son was born in 2001, making him 13 now, give or take. So if you were imagining the chancellor rolling around on the floor with a cute toddler and demonstrating how to make a three-dimensional Laffer curve out of bricks, think again. Picture him, instead, trying to make awkward conversation with a towering, grunting adolescent who is demanding to know if the old posh bloke prefers Call of Duty: Ghosts or Fallout 3, and then firing off a look of withering contempt if he proffers the wrong answer. Babysitting my foot.

Nonetheless, this little bit of tabloid myth-making has some purchase, as they say. It fits nicely with a prevailing narrative about our contemporary ruling class. They may be (overwhelmingly) expensively educated, white men in grey suits, but they are the new domestic egalitarians, mucking in at home, changing nappies, throwing breakfast on to the table for their kids while they and their independently wealthy, career-making wives slip into their his 'n' hers Paul Smiths for the day.

Just how honest this image might be is a moot point. One suspects a small army of minimum wage cleaners, nannies and au pairs is stitching together the seams of the Lean In establishment and its illusions. However there is a message in the mythology. Our politicians (including the millionaires and direct descendants of royalty) are keen to portray themselves as ordinary people with ordinary families, and in this picture, ordinary dads are hands-on and involved. It seems the first public demand on a male politician is that he be a good dad, despite a parliamentary system and working practices for MPs that many still find incompatible with active parenting.

All this may be a key message among the ABC1 swing voters in marginal seats, but I'm not sure the rest of our culture has quite caught up yet. We live in a society that has community mother and toddler groups, primary schools that address letters to mums, right up to the family court system that still generally presumes mothers, not fathers, to be the more natural carer of a child after separation, so it is a society riven with assumptions about parenting. Some of these are imposed by policy and structure, but many simply grow out of our collective attitudes, beliefs and behaviours.

This week Duncan Fisher, a founder of the Fatherhood Institute, launches a new website called mumsanddadsnet. The name is a deliberate echo – possibly a sly dig – at the mother of all parenting websites, Mumsnet. The new site's mission statement points out that the online conversation about parenting is divided between mums, on the one hand, and dads (mostly separated ones), on the other. "We are stuck in a paradigm – that mums do parenting and dads help. This idea might suit a lot of people, but it's plain wrong."

Justine Roberts, the founder of Mumsnet, has pointed out, correctly I'm sure, that its Dads section is already the largest specialist area for fathers in British cyberspace. That in itself, perhaps, is the main problem. Dads are tacking themselves on to the efforts of mums as a sideshow and an afterthought. Roberts went on to say "I agree with the principle; it would be nice if dads wanted to identify as dads. But they don't. They identify as men who talk about parenting."

As a proudly self-identifying dad who spends a lot of time talking to other self-identifying dads, this strikes me not only as utterly wrong, but a profoundly harmful projection. The fact that the chief executive of Mumsnet would come out with such an alienating pronouncement represents game, set and match to Duncan Fisher, I think, in the debate as to whether we need less compartmentalisation and factionalism in debates about parenting.

The story of two senior male politicians helping each other out with the babysitting may be a comforting fiction, but there is no doubt that all parents and carers, whatever their gender or role, need a degree of support and assistance. It is too early to guess whether the latest online initiative will provide a successful, gender-neutral alternative to the might of Mumsnet, but the effort at least should be applauded.